(1).已知向量a=(2cosx,cosx+sinx),向量b=(sinx,cosx-sinx)
而函数f(x)=向量a*向量b,
所以,
f(x)=2sinxcosx+(cosx+sinx)(cosx-sinx)
=2sinxcosx+cos²x-sin²x
=sin2x+cos2x
=√2sin(2x+π/4)
f(x)的图像对称中心(-π/8+kπ/2,0),k∈Z
f(x)的图像对称轴方程为x=π/8+kπ/2,k∈Z
(2).对任意x∈[0,π/2],有f(x)＜m²+m+√2-2恒成立,
由x∈[0,π/2],2x+π/4∈[π/4,π],
f(x)max=f(π/8)=√2
由题知,f(x)＜m²+m+√2-2恒成立,
所以,√2＜m²+m+√2-2
m²+m-2＞0
得m∈(-∞,-2)∪(1,+∞)